ACKNOWLEDGMENTS This is a humble effort to express our sincere gratitude towards those who have guided and helped us to complete this project. A project reported is major milestone during the study period of a student. We could have faced many problems but our teachers’ kind response to our needs and requirement, their patient approach and their positive criticism helped us in making our project. Very warm thanks to our project-in-charge “SWATI AGGRAWAL” with her support and constant encouragement AND LPU LIBRARY it was not very easy without whose support to finish our project. With the motivation of our parent it was very easy to finish our project successfully and satisfactorily in short span of time.

Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example 4x2-2xy+3y2

is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory q(x)=ax2 q(x,y)=ax2+bxy+ay2 q(x,y,z)=ax2+by2+cz2+dxy+ex2+fy2 , where a,…,f are the coefficients.Note that quadratic functions, such as ax2+bx+c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the p-adic integers ZpBinary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology.

Using homogeneous coordinates, a non-zero quadratic form in n variables defines an (n−2)dimensional quadric in the (n−1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections.

History

The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form x2 + y2, where are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.

In 628, the Indian mathematician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many other things, a study of equations of the form x2 − ny2 = c. In particular he considered what is now called Pell's equation, x2 − ny2 = 1, and found a method for its solution.n Europe this problem was studied by Brouncker, Euler and Lagrange.

In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.

POSITIVE DEFINITE QUADRATIC FORM: An n × n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries ( ), where zT denotes the transpose of z. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z. The quantity z*Mz is always real because M is a Hermitian matrix A necessary and sufficient condition for a real quadratic form X’AX to be positive definite is that the leading principle minorsx of the matrix A of the form are all positive.

LEMMA:
THE DETERMINANT OF THE MATRIX OF A DEFINITE FORM OS POSITIVE.

IF X’AX to be positive definite form ,,then there exists a real non singular matrix such that P’AP=I |P’AP|=|I| |P’||A||P|=I |A|=I/|P2| THUS |A| IS POSITIVE

THE CONDITION IS NECESSARY:
Let the form X’AX be positive definite .let s,be natural number less than or equal to n Putting xs+1=0.....xn==0

In the positive definite form X’AX we obtain a positive definite quadratic form in,s, variables x1.....xs,the determinant of whose matrix is the leading principle minor of order sof a.Thus, by lemma,every leading principle minor matrix of appositive definite quadratic form is positive. The result is true for quadratic forms in asingle variable,since A11x12 Is positive definite quadratic forms in m variables Consider any quadratic form in (m+1) variables with corresponding symmetric matrix such that the leading principle minors of its matrix s,are all positive.we partition s as follow

S= The leading principle minors of b are all positive ,the leading principle minors of b and the determinant|s| are all positive and |p| is not equal to zero. As the theorm is assumed to be true for quadratic forms in m variables,there exist a non singular matric p of order m such that P’BP=Im Let C be a column matrix to be shortly determined in a Suitable manner. We have = With C determined as above we have P’BC+P’B1=0, C’BP+B1’P=0,C’BC+C’B1=0 THUS S = = Taking determinants ,we obtain |P’||S||P|=|I||B1’C+K|=B1’C+K AS |S|is positive and |p| is not equal to 0,we see that B1’C+K is also positive B1’C +K=β2 Where β is real.Thus S = Pre multiplying and post multiplying with
-1

S Writing Q= Q’=
-1 -1

=IM+1

WE OBTAIN Q’SQ=IM+1 AS S IS CONGRUENT TO IM+1 ,WE SEE THAT CORRESPONDING QUADRATIC FORM IN (M+1) IS POSITIVE DEFINITE DEFINITE ,SEMI DEFINITE AND INDEFINITE REAL QUADRATIC FORM Let X’AX be real form in n variables x1,x2,x3....... With rank r and index p. The form is said to be 1: Positive definite form: if r=p=n if the rank and index of quadratic form are equal. 2: Negative definite form: If r=n,p=0,the index is zero 3: Positive semi definite form: If r<n,p=r that is the real quadratic form is singular 4: Negative semi definite : If r<n,p=0 that is index is zero and the form is reduced by non singular transformation x=py

EXAMPLES OF QUADRATIC FORM WHICH IS POSITVE DEFINITE 1: PROVE THAT THE QUADRATIC FORM 6x12+ 3x22+14x32+4x2x3+18x3x1+4x1x2 is positive definite.

THE MATRIX OF THE QUADRATIC FORM IS

A=

=DETERMINANT=1

HENCE QUADRATIC FORM IS POSITIVE DEFINITE 2:PROVE THAT THE QUADRATIC FORM 4x2+9y2+2z2+8yz+6zx+6xy IS NOT POSITIVE DEFINITE THE MATRIX OF THE QUADRATIC FORM IS

A=

=DETERMINANT=-19

SINCE DETERMINANT IS NEGATIVE,THE QUADRATIC FORM IS NOT POSITIVE Characterizations Let M be an n × n Hermitian matrix. The following properties are equivalent to M being positive definite: 1 All eigenvalues λi of M are positive. Recall that any Hermitian M, by the spectral theorem, . may be regarded as a real diagonal matrix D that has been re-expressed in some new coordinate system (i.e., M = P − 1DP for some unitary matrix P whose rows are orthonormal eigenvectors of M, forming a basis). So this characterization means that M is positive definite if and only if the diagonal elements of D (the eigenvalues) are all positive. In other words, in the basis consisting of the eigenvectors of M, the action of M is component-wise multiplication with a (fixed) element in Cn with positive entries[clarification needed]. 2 The sesquilinear form . `(x,y) = y*m(x) defines an inner product on Cn. (In fact, every inner product on Cn arises in this fashion from a Hermitian positive definite matrix.) 3 M is the Gram matrix of some collection of linearly independent vectors . X1.......XN BELONGS TO CK for some k. That is, M satisfies: MIJ=MI *MJ The vectors xi may optionally be restricted to fall in Cn. In other words, M is of the form A*A where A is not necessarily square but must be injective in general. 4 All the following matrices have a positive determinant (Sylvester's criterion): . the upper left 1-by-1 corner of M the upper left 2-by-2 corner of M the upper left 3-by-3 corner of M

... M itself In other words, all of the leading principal minors are positive. For positive semidefinite matrices, all principal minors have to be non-negative. The leading principal minors alone do not imply positive semidefiniteness, as can be seen from the example

5 There exists a unique lower triangular matrix L, with strictly positive diagonal elements, . that allows the factorization of M into M = LL * . where L * is the conjugate transpose of L. This factorization is called Cholesky decomposition.